This approach is generally referred to as covariance truncation.
(5), as in the remainder of this article, denotes an estimator of the precision matrix, as opposed to the inverse of an estimator of the covariance matrix. Where is the diagonalized factorization of with its matrix of eigenvectors where with is its matrix of eigenvalues and where only the k leading eigenvectors of the empirical covariance are retained ( ).
Practically, the results of the inference and, in particular, the magnitude of the uncertainty range on the regression coefficients β, determine whether these fingerprints are present in the observations and whether or not the observed change is attributable to a given cause. The regressors -or fingerprints-consist here of spatial, temporal, or space–time patterns of response to external forcings as anticipated by one or several climate models, whereas internal climate variability ε is treated as a multivariate Gaussian noise with mean zero and covariance. From a methodological standpoint, D&A studies involve a range of methods (e.g., correlation comparison, regression-based decomposition, and Bayesian inference) but are frequently based on linear regression methods referred to as optimal fingerprinting, whereby an observed climate change is regarded as a linear combination of p externally forced signals added to an internal climate variability term ( Hegerl and Zwiers 2011). Over the past two decades, the results produced in this area have been instrumental in delivering far-reaching statements and in raising awareness of anthropogenic climate change. The method is illustrated on twentieth-century precipitation and surface temperature, suggesting a potentially high informational benefit of using the raw, nondimension-reduced data in detection and attribution (D&A), provided model error is appropriately built into the inference.Īn important goal of climate research is to determine the causes of past global warming in general and the responsibility of human activity in particular this question has thus emerged as a research topic known as detection and attribution (D&A). both estimation error and accuracy of confidence intervals and also highlight the need for further improvements regarding the latter. Results on simulated data show improved performance compared to existing methods w.r.t. Further, it is shown that preliminary dimension reduction is not required for implementability and that computational issues associated to using the raw, high-dimensional, spatiotemporal data can be resolved quite easily. Point estimates and confidence intervals follow from the integrated likelihood. This allows for the introduction of regularization assumptions. The unknown covariance is treated as a nuisance parameter that is eliminated by integration. The suggested approach is based on a single-piece statistical model that represents both linear regression and control runs. It is argued that such a compartmentalized treatment presents several issues an integrated method is thus introduced to address them. These methods used to involve three independent steps: preliminary reduction of the dimension of the data, estimation of the covariance associated to internal climate variability, and, finally, linear regression inference with associated uncertainty assessment. The present paper introduces and illustrates methodological developments intended for so-called optimal fingerprinting methods, which are of frequent use in detection and attribution studies.
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